My father and elder brother and even myself some times went to Synague, see I dont even know how to spell it, and one of my brothers and I are proud to be Atheist. Trust your children as if they are given the right information by their parents then they will see that religion is complicated and cant all be true. Let them ask any questions .
I would allow my child to be alone with a heavily religious family member, as long as I trust them to look after my child. For example, kindly and religious granny with cookies is fine, creepy uncle who also happens to be religious is not.
Kids are resilient. It takes quite a lot of effort to brainwash someone. A couple of hours hearing about hell from Aunty Sue is unlikely to scar the child for life. It also leads to cool discussions about the problems with religious concepts, like heaven and hell.
Bottom line, for me, is that exposure to irrational beliefs is a great opportunity to teach your child rational thinking. Not preaching and bitching about how wrong gran's beliefs are, but asking questions, like "Oh, is that what she said? What do you think about that?" It's allowing questions that makes the difference. I was told "virgin birth, living inside a whale, that's the way it is, no asking questions, on yer knees, pray!" That's what screws you up, not the stories themselves.
I would surmise that every member of TA had religious relatives who tried to persuade them to accept Jesus as their lord and savior sometime in their lives, and we seem to have weathered it. Teach you children well, they will be what they will be.
I agree with some of the others here, in that I'd recommend telling your family members that you don't want to brainwash your kids, but that solitary exposure to religious family doesn't need to be harmful. Why should we worry about the effect of being around religious family has any more than being around family members who think Santa is real, or the Tooth Fairy? In my opinion, you just sit down with them and explain that some people believe in odd stuff that isn't real.
I haven't read the responses yet, but yes, I would let a religious family member or friend take care of my kids. Why?
1) All of my family members are very devoutly faithful. So are most of my close friends.
2) I'd want them to feel comfortable leaving their children with me. Why should there be a double standard?
3) I'd make it damn clear what they could or could not teach my kid. I can't control it, but they'd know that in order to spend time with my kid, they'd have to agree to my conditions as a mother.